Tavola degli integrali indefiniti di funzioni razionali: differenze tra le versioni

Questa pagina contiene una '''tavola di [[integrale indefinito|integrali indefiniti]] di funzioni razionali'''. C denota una costante arbitraria di integrazione che ha senso specificare solo in relazione a una specificazione del valore dell'integrale in qualche punto.

: <math>\int (ax + b)^n dx = \frac{(ax + b)^{n+1}}{a(n + 1)}+C \qquad\mbox{(per } n\neq -1\mbox{)}</math>

: <math>\int x^{n-1}(ax^{n} + b)^{c}\;dx = \frac{(ax^{n} + b)^{c+1}}{na(c + 1)}+C </math>

: <math>\int\frac{dx}{ax + b} = \frac{1}{a}\ln\left|ax + b\right|+C</math>

: <math>\int x(ax + b)^n dx = \frac{a(n + 1)x - b}{a^2(n + 1)(n + 2)} (ax + b)^{n+1} +C\qquad\mbox{(per }n \not\in \{-1, -2\}\mbox{)}</math>

: <math>\int\frac{x\;dx}{ax + b} = \frac{x}{a} - \frac{b}{a^2}\log\left|ax + b\right|+C</math>

: <math>\int\frac{x\;dx}{(ax + b)^2} = \frac{b}{a^2(ax + b)} + \frac{1}{a^2}\log\left|ax + b\right|+C</math>

: <math>\int\frac{x\;dx}{(ax + b)^n} = \frac{a(1 - n)x - b}{a^2(n - 1)(n - 2)(ax + b)^{n-1}}+C \qquad\mbox{(per } n\not\in \{-1, -2\}\mbox{)}</math>

: <math>\int\frac{x^2\;dx}{ax + b} = \frac{1}{a^3}\left[\frac{(ax + b)^2}{2} - 2b(ax + b) + b^2\log\left|ax + b\right|\right]+C</math>

: <math>\int\frac{x^2\;dx}{(ax + b)^2} = \frac{1}{a^3}\left(ax + b - 2b\log\left|ax + b\right| - \frac{b^2}{ax + b}\right)+C</math>

: <math>\int\frac{x^2\;dx}{(ax + b)^3} = \frac{1}{a^3}\left[\log\left|ax + b\right| + \frac{2b}{ax + b} - \frac{b^2}{2(ax + b)^2}\right]+C</math>

: <math>\int\frac{x^2\;dx}{(ax + b)^n} = \frac{1}{a^3}\left[-\frac{1}{(n- 3)(ax + b)^{n-3}} + \frac{2b}{(n-2)(a + b)^{n-2}} - \frac{b^2}{(n - 1)(ax + b)^{n-1}}\right]+C \qquad\mbox{(per } n\not\in \{1, 2, 3\}\mbox{)}</math>

: <math>\int\frac{dx}{x(ax + b)} = -\frac{1}{b}\log\left|\frac{ax+b}{x}\right|+C</math>

: <math>\int\frac{dx}{x^2(ax+b)} = -\frac{1}{bx} + \frac{a}{b^2}\log\left|\frac{ax+b}{x}\right|+C</math>

: <math>\int\frac{dx}{x^2(ax+b)^2} = -a\left[\frac{1}{b^2(ax+b)} + \frac{1}{ab^2x} - \frac{2}{b^3}\log\left|\frac{ax+b}{x}\right|\right]+C</math>

: <math>\int\frac{dx}{x^2+a^2} = \frac{1}{a}\arctan\frac{x}{a}+C</math>

: <math>\int\frac{dx}{x^2-a^2} = -\frac{1}{a}\,\mathrm{settanh}\frac{x}{a} = \frac{1}{2a}\log\frac{a-x}{a+x}+C \qquad\mbox{(per }|x| < |a|\mbox{)}</math>

: <math>\int\frac{dx}{x^2-a^2} = -\frac{1}{a}\,\mathrm{settcoth}\frac{x}{a} = \frac{1}{2a}\log\frac{x-a}{x+a}+C \qquad\mbox{(per }|x| > |a|\mbox{)}</math>

: <math>\int\frac{dx}{ax^2+bx+c} = \frac{2}{\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}}+C \qquad\mbox{(per }4ac-b^2>0\mbox{)}</math>

: <math>\int\frac{dx}{ax^2+bx+c} = -\frac{2}{2ax+b}+C \qquad\mbox{(per }4ac-b^2=0\mbox{)}</math>

: <math>\int\frac{dx}{ax^2+bx+c} = -\frac{2}{\sqrt{b^2-4ac}}\,\mathrm{settanh}\frac{2ax+b}{\sqrt{b^2-4ac}} = \frac{1}{\sqrt{b^2-4ac}}\log\left|\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right|+C \qquad\mbox{(per }4ac-b^2<0\mbox{)}</math>

: <math>\int\frac{mx+n}{ax^2+bx+c}dx = \frac{m}{2a}\ln\left|ax^2+bx+c\right|+\frac{2an-bm}{a\sqrt{4ac-b^2}}\arctan\frac{2ax+b}{\sqrt{4ac-b^2}} +C\qquad\mbox{(per }4ac-b^2>0\mbox{)}</math>

: <math>\int\frac{mx+n}{ax^2+bx+c}dx = \frac{m}{2a}\ln\left|ax^2+bx+c\right|+\frac{2an-bm}{a\sqrt{b^2-4ac}}\,\mathrm{setttanh}\frac{2ax+b}{\sqrt{b^2-4ac}} +C\qquad\mbox{(per }4ac-b^2<0\mbox{)}</math>

+ \frac{1}{4\sqrt{2}} \left[ \log |x^2+\sqrt 2 x +1| - \log |x^2-\sqrt 2 x +1| \right]+C

: <math>\int \frac{dx}{x^{2^n} + 1} = \sum_{k=1}^{2^{n-1}} \left\{ \frac{1}{2^{n-1}} \sin\frac{(2k -1) \pi}{2^n} \cdot \arctan\left[ \left(x - \cos\frac{(2k -1) \pi}{2^n} \right ) \csc\frac{(2k -1) \pi}{2^n} \right] - \frac{1}{2^n} \cos\frac{(2k -1) \pi}{2^n} \cdot \log \left| x^2 - 2 x \cos\frac{(2k -1) \pi}{2^n} + 1 \right| \right\} +C </math>